"Plus" and "Minus" States of Linear Molecules Frank L. Pilar University of New Hampshire Durham. NH 03824 Many chemistry courses require students to predict the soectroscooic term svmbols of linear molecules from molecu'lar-orbitai e~ectroniEconfigurations. The spectroscopic term symbols identifying electronic states of linear molecules are based on the hond-axis component of the total electronic orbital aneular momentum. This comwonent-which is analogous to M,, of atoms but is used in thk sense of L-is given the symbol A. For linear molecules with a center of symmetry (Dm,,point group), the spectroscopic term symbol has the general form 2s+lli\l+ur"re where IA1 is designated by 2,Il,A, a,. . . when its value (in units of h ) is O,1,2,3,. ..When 1A1 # 0, the i right-hand superscript is not used, and for linear molecules without a center of symmetry (C,, point group), the u , g, subscript is not applicable. Although students master the prediction procedures quite easily-largely by analogy to the previously-learned Russell-Saunders case in atoms-- one asoect of the molecular term symbols typically remains puzzling: the precise significance of the "olus" and "minus" suoerscriots used for 2: states. T h e four qu&tions most often asked by students are
.
1) Why do electron configurations having only one electron in nu, or na.. molecular orbitals (in addition to a comoletelv . . filled shell) always turn out to be ZXt: i.e.. why does 2X-not appear?
2) Why do closed-shell electron configurations always lead Ma "plus" state? states arising from a half-filled dou3) Why are the I & and bly-degenerate shell always "plus" and "minus." respectively? Rz with (la,S2and 0 2 with (1rRS4arecases in point. 4 ) Why are "plus" and "minus" not used for Il,A, a,.. .states? Some texts do make some attempt to justify the f designations, e.g., those by Slateri and by Pilar.2 However, both texts leave enoueh eaos . in the details of the reasonine that onlv students unusually gifted in abstract reasoning a b k t y will de convinced. I t is the intent of the oresent . oawer . to suoolv .. . the details needed for giving complete answers to the four questions above. The f designations arise from the axial symmetry cbaracteristic of the C,., and D-h. . point proups. Any obiect having this axial symmetry can be divided i n k two identical halves by slicing it parallel to its principal axis of symmetry (C,). This illustrates a mirror plane symmetry element which is called a,.. Because of the C, axial svmmetrv there are an infinite numher of equivalent ways of making the division, i.e., there arean infinitenumber of mirror planes; this is the m a , column in the C,, and D,h character tables. Any mathematical function of overall C,, or D,h symmetry wil be designated (+) if it transforms as the At irreducible representation (also called ZC)of the C,, point group or the Atr or A,, irreducible representations (also called 2: and 2:) of the D,h point group. The character of a a , symmetry operation for these irreducible representations is +l. If the function transforms as A O - ) of C,, or AW(&3 or Az.(Z;) of D,h, it is called (-) and the character of the a, operation is -1.
.,
Slater. J. C., "Quantum Theory of Molecules and Solids," McGraw-Hill Book Co.. Inc.. New York, 1963. pp. 117-120. Pilar. F. L.. "Elementary Quantum Chemistry." McGraw-Hill Book CO.. Inc.. New York. 1968, pp. 504-506. 758
Journal of Chemical Education
Now consider a linear molecule with the molecular-orbital electron configuration
where the first N molecular orbitals (each doublv occuwied) constitute a closed-shell, and where d, is a doubly:degenerate molecular orbital. From this electron confiruration one can coitstruct wavefunrtims hirh nrr simultn~eouseigeniunctions of the elrctronir Hamiltonian H (In the h r n - O p p e n h ~ i m e rapproximationl and various other operat!lrs which rommute with I?. M'eshall cmsider the operators I,,. S'.., .ql. and !'-all of which commute u,ith the Hamiltcmian ol a linear molecule. The operator i, describes the quantization of orbital angular momentum about the bond axis; its eigenvalue equations are of the form
..
where X = 0, f1, f2,. .The quantum number X is analogous to them, quantum number of atoms. Nondegenerate molecular orbitals have X = 0 and are designated a. Some examples for homonuclear diatomic molecules are a n l s , a i l s , an2s, a 3 , an2p, and a;2p (also written l a n , la,,, 2a,, 2a,, 3an, and 3a,,). Doubly degenerate molecular orbitals have A z 0 and are called n,& d,. . ..when IXI = 1,2,3,. . ..The X-value of a molecular orbital formed by the LCAO approximation comes from the mi values of the atomic orbitals used to construct it. Thus, all a molecular orbitals are formed from atomic orbitals such as Is, 2s, 2p,,, etc., all having m, = 0. Similarly, x molecular orbitals are made up from atomic orbitals such as 2p l,3dl,. . . (all having mr = 1) or 2p-I, 3d-I,. . . (all having mr = -1). The operator S? represents the scalar square of the total spin angular momentum, and gZ represents the bond-axis component of that angular momentum. The reader is referred to any of the standard texts on quantum chemistry for a discussion of these operators. Their eigenvalue equations are
S = 0. I, 2.. . . (even number of electrons) =
1/2.3/2,5/2,. . . (odd number of electrons)
&$ = Msh*
T h e operator P is related to the ma,, symmetry operations of the C,, and D,h point groups. If $i(p) represents nondegnerate simultaneous eigenfunctions of and f', then the eigenvalue equations of may be written
a
P*,(v)= **;(+I
(4)
where, as shown in Figure 1, p i s the spherical polar coordinate of an electron rotating about the bond axis. This means that the eigenvalues of are -1 and +1. T h e operator P has the effect of changing the angle p to -q, i.e., T h e + I eigenvaluearises when $,(-6)= thisiscall is called a "plus" state IZ'!, 21, Xi).T h e -I eigenvalue arlses when
+;(-p) = -+;(p) and leads to a "minus" state (2;. Z,; 2). Thus. we see that (+I and (-)are used to distinguish different nondegenerate states. Now consider two doubly degenerate eigenfunctions of fl (but not necessarily o f p ) which we designate as u(p) and u b ) . Operating on these functions with P produces
f converts one degenerate eigenfunction of fl into the i.e., ' other. This means that we can take the sum and difference of ~ ( pand ) u(v)to produce simultaneous eigenfunctions of both I? and P. P(u + u) = U + u
P(u - u) = -(u - u )
(7)
These also have the eigenvalues +I and -1, respectively. However. douhlv.deeenerate states are not labelled with the . f symhois; such a designation is unnecessary since the two states have the same enerzv. It should he noted that ihechararterof the m a , , operation is always zero for the douhly degenerate states. This occurs because the trace of the matrix represenmtion of the operator I' for doubly degenerate irreducll)le representations is zer-the sum of the -1 and +1 eigenvalues. If we let &+ and 4- represent the two doubly degenerate molecular orbitals in the electron configuration (1) (one with A = +n and one with A = -n, where n = 1.2,. . .), then six different determinantal functions may be constructed. These are
(doublydegenerate) J.['l21hl),] = Dsur DI; All but D gand Dsare alsoeigenfunctions of P. We shall return to this point later. The notation 1(21AI), is exemplified by 02(1r,)') where IAI = 1 leading t o a 'A, state; for a molecule with two electrons in n6, or n6, molecular orbitals, IAI = 2 and a doubly degenerate IT, state ( I A I = 4) arises. Exceot for the f oar& of the term svmbols. all of the above can he deduced rquiie readily in the same way one establishes that carinn ls'2&o'leads tot hestates IS, 'I), and ' I 1 ;simply put twoelectrons into@+and 4- inall possible ways with&t violatine the exclusion . orincale. . This procedure is illustrated in Figure 2. If the determinants D Iand D2 are expanded by factoring out the elements of -the ficst c o k m n , @ , ( l ) n ( l ) b ~ (2)n(2) . . b(N 2 M N 2), until only 2 X 2 determinants containing @+ and 4- remain, one obtains a sum of terms each having the general form ~~
~~
.
.
+
+
For the singlet state (lower signs)
For the triplet state (upper signs),
In the above, b; (without a prime) represents a column in the determinant in which each electron (1,2,3,. . . N 2) has a spin, and @; (with a prime) correspondingly represents @ spin for each electron. Each of these six determinants is a simultaneous eigenfunction of I?, L,, and 9,but not necessarily of S 2 and P. If the sum and difference of D sand D4are used to replace D Rand D4,a11 become eigenfunctions of 3 2 . These represent the electronic states given below
+
Fbure 1. Spherical palar coordinate system tw an electron In a diatomic mle~uleAB. T M angle q describes motion about the bond axis ( 2 axis).
T h e term F(2N) contains permutations of the closed-shell orbitals gl, 42,. . .b,v among 2N of the 2N 2 electrons. In general, there will be
+
terms in F(2N). Each such term is a permutation of 2N of the 2N 2 electrons, or, alternatively, each determinant multiplying F(2N) represents a different pair (pp) of electrons. For terms(N example,R2: (la,)2(1~.)~(2o,)2(2a,)~(ln,)~as45 = 4) and 0 2 : (10g~2(lau)2(2a~)2(2uu)2(l~u)4(3u~~Y(1r,)2 has 120 terms ( N = 7). Equations (10) through (13) follow from the antisymmetry principle and can he deduced from a knowlege of the simple cases of the IS and ?S states arising from the ls2s electronic configuration of the helium atom, a topic customarily discussed in some depth in elementary courses in quantum
+
Figwe 2. DBduction of me spectroscopic states ol E2: llap)2~ls.)2(2a,)z(2o,)2(i~P.The multiplichier. A-values, and v+ characters are obviars, but me 5 designationsare mysterious at this stage. Volume 58 Number t o
October 1981
759
theory. Except for the fact that 1s and 2s are not degenerate (a fact irrelevant to the results), the ls2s configuration is a special case ( N = 0) of the electron configuration (1). The helium wavefunctions are $llS) = Ills2.v'~- llr'2sI)Ifi
Since the singlet D;,- D I contains m+(p)d-(a) + +-(ph,h+(v), operating with f'on each of the ( N + 1)12N 1) terms leaves I):, - /I4 unchanged. Cnnsequently, this is a 1x1 state. On the other hand, the tripletsI)~.Il~. and D:I + I1,each which changes sign on opcontain Q+(p!+-(r,) - h-(p)++(~') eration with I? Consequently, these are components ofa:'XJ state. Since P changes d,+(p)+-(v) to 4-(p)4+(1,)and vice-versa. it is apparent that
+
+
Thus, D;, Dl; and D i - D,; are eigenfimctions of /? When I A1
where the singlet spatial function is symmetric with respect to interchange of spatial ccwrdinates of the two electrons and the corresponding spin function is antisymmetric with respect to the interchange ul'spin coordinates of the two electrons. For the triplet wavefunctinns, the opposite symmetries hold: the spatial function is antisymmetric and the spin functions are symmetric. It follows for the general case ( N > 0) that each pair of electrons in any singlet or triplet determinantal wavefunction must appear in the detenninantal expansion with spatial and spin parts analogous to the simple cases shown in the helium atom; thus, the forms of the terms represented by eqns. (10)-(13) are readily justified. T h e operator P affects ++ and 4- as follows,
Pa+ = m(15) Pa- = r+ because 4+ contains e i l " ~and 6- contains e-'lAlr. For example, in R2
where A and B refer to the two different atoms of the molecule and Iindicates that the 2 p orbitals are perpendicular to the bond axis. If we use one set of 2p orbitals (those with mc = X = +1 and containing e'.,) to form 4+, then the other set (those with mc = h = -1 and containing e-'"1 forms $-. Consequently, the operator P merely changes X = + I to A = -1 and vice- versa.
760
Journal of Chemical Education
= 1,these lead tothedouhlydegenerate 'A, state (1.11= 21AI = 9) A , .
The mdecular d ~ i l a l s o f nand ~ a,, t q r nre I~nearcomllinations 01 atomlc s-twhitals im/ = 0)and. nmiequentlv, have X = 0. Thus. a single electron in a, or n,, prnd~k!esZ Z ' f and %~,respectively,since Pn, = a, and Pn,, = a,,. A closed-shell such as (a,,)"or (T,).' produces '2: since applying 15 to a function such as 9+(1)4+(2)*-(:X),b-(4) produces a permutation of electron coordinates already present in the determinnntal expansion, Consequently, the operator P merely reshuffles the nrder in which the permutations nccur and leaves i he total wavefunction unchanged: this produces a "plus" state. A short qualitative summary is now in order. There is only one way an electron can have zero orhital angular momentum (A = 0) and thiscan occur in eithera (+) state or a (-)state. Furthermore, these two states will have different energies. Classically, zero angular momentum means that the electron is motionless, hut wave-particle duality and the uncertainty principle replace this with a probability distribntiun which hascylindrical symmetry and thusa net orbital angular momentum ol'zern. When thestate is doubly degenerate, thenet motion of the electron can be either clockwise ( X > 0) or anticlockwise ( A < 0).Since both situations can have either I t ) or (-) eigenvalues of P (or a mixture of both), and since both states are of eaual enerev. -. the (fI designation is of no value for classifying states. The author wishes to thank an anonymous reviewer for suggesting some minor hut important changes in the original manuscript.
